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Triple Integrals : Changing the Order of Integration and Finding the Volume of a Wedge

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1. Using the integral ∫-1-->1 ∫x^2-->1 ∫0-->1-y dz dy dx
a) Sketch the region of integration.
Write the integral as an equivalent iterated integral in the order:
b) dy dz dx
c) dx dz dy
d) dz dx dy

2. Find the volume of a wedge cut from the cylinder x^2 +y^2 =1 by planes z=-y and z=0.
Please show me the process too.

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Triple Integrals, Changing the Order of Integration and Finding the Volume of a Wedge are investigated. The solution is detailed and well presented.

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The volume of integration can be seen in the figure below:

As you can see the z coordinate is bound between the plane z=0 on top and the line z=1-y on the top.
The y coordinate is bound by the plane y=1 from the right and the parabola y=x2 from the left.
x is simply bound between the planes x=±1 (front and back)

Conversion to dzdydx

We need to express y as functions of z and x, the limits of z as a function of x and the limits of x as constants.

Since the original limits of x are already constants, they stay the same.

The relation between y and z can be written as:

This is the upper limit of y. The original lower limit of y is x2. Since x is the last variable in the integration order, this can ...

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